Convexity and Haar null sets

Abstract

It is shown that for every closed, convex and nowhere dense subset C C of a superreflexive Banach space X X there exists a Radon probability measure μ \mu on X X so that μ ( C + x ) = 0 \mu (C+x)=0 for all x ∈ X x\in X . In particular, closed, convex, nowhere dense sets in separable superreflexive Banach spaces are Haar null. This is unlike the situation in separable nonreflexive Banach spaces, where there always exists a closed convex nowhere dense set which is not Haar null.